FOM: Categorical vs set-theoretical foundations

The idea that *any* formal axiomatic theory can serve as a
foundation for mathematics is profoundly mistaken - I would go so far
as to say that it is logically absurd. This applies as much to ZFC as
to the theory of a topos with whatever additional axioms you like. Let
me explain.
When you employ the modern axiomatic method your axioms,
whether formal or informal, together define the class of mathematical
structures in which they all hold true: that is their *logical*
function. Thus to say that a theory is a *formal* axiomatic theory is
to identify the truths of that theory - its theorems - with the set
composed of all those sentences in the language of that theory that
hold in all structures that model the axioms of the theory.
Thus the formal theory of real numbers (the theory of complete
ordered fields) is a *second order* theory, and its truths are the
second order logical consequences of the axioms for a complete ordered
field. Similarly, the formal theory of groups is a *first order*
theory whose truths are the first order logical consequences of the
group axioms. The same applies to the first order theory of toposes (or
of toposes with a natural number object, or whatever) and, indeed, to
the first order theory of sets, ZFC.
Now with second order axiomatic theories, and with non-formal
ones, you have no choice but to talk about the structures that are
their models, about morphisms between such structures, about
isomorphism classes, etc.. And all this talk has to have real
content: otherwise you can neither explain what you are doing or
justify it. Indeed, you cannot even make sense of it, and if you try to
formalize that discourse in a formal topos theory or set theory or
whatever, you will simply be confronted with the problem of explaining
how *that* theory works, and why it is justified, and that is a task of
the same sort as your original one.
Now it might be thought that in the case of 1st order axiomatic
theories you could avoid talking about models since first order logical
consequence can be defined syntactically in terms of formal proofs from
the axioms rather than semantically in terms of truth in all models of
the axioms. But such a view is intellectually incoherent, and the
reason for this is simple and quite straightforward: you cannot
*justify* a system of proof procedures without using the the general
notion of a model.
Surely it is obvious that you have to justify a system of
formal logical proof. Why should logical consequence be *defined* to be
"formally derivable in just *this* system of formal rules and logical
axioms"? To be sure, each formal derivation clearly witnesses a logical
consequence (because the rules are *sound*). But how can you be sure
that nothing has been left out (that is to say, how do you know that
your system of rules, taken as a whole, is *complete*)? Although we
have intuitions of the *soundness* of individual rules and logical
axioms, we clearly do not have intuitions of the completeness of whole
*systems* of axioms and rules. Goedel's Completeness Theorem is, after,
a deep and indispensable result in first order logic.
So in order to make sense of what you are doing, even in first
order logic, you must take the semantic notion of logical consequence
as logically prior to, and more fundamental than, the definition of
formal derivation. And this clearly means that any attempt to use a
formal axiomatic theory as a logical foundation for the whole of
mathematics will inevitably involve you in a gross and obvious vicious
circle. For the chief requirement of such a theory would be to explain,
and to justify, the axiomatic method itself.
Of course it is possible to maintain that a particular first
order theory (e.g.. ZFC) contains a formal analogue of any proof that
mathematicians at large are likely to accept as valid. In those
circumstances you can sometimes conclude that a proposition whose formal
analogue is formally undecided by the axioms (e.g. CH) is beyond the
reach of present day mathematics. But such conclusions are not always
warranted (e.g the consistency of ZFC is independent of ZFC, but any
one who accepts ZFC is highly likely to accept its formal consistency
too). This, however, is a far cry from saying that the formal first
order theory ZFC provides a foundation for mathematics.
But that is how Vaughn Pratt understands "set-theoretical"
foundations:
"To avoid misunderstanding it should be said that ZF itself does not
start with a preconceived notion of collection and go from there. ZF
assumes only the language of first order logic with equality and an
uninterpreted binary relation (membership). In fact, there is no a
priori assumption at all about the universe being described by the
axioms, other than the traditional (but unnecessary) assumption that it
is non-empty. ZF weaves the entire notion of set from the whole cloth
of first order logic."
Apparently Harvey Friedman agrees with Pratt, at least on the
foundational use of formal first order axiomatic theories. Why else
should he produce a set of axioms and then challenge the category
theorists to match them in point of elegance, economy, etc.?
But if Pratt's description of ZF is intended to be an account
of "set theoretical foundations" then he is just wrong. And Friedman's
challenge is irrelevant to the point at issue.
To provide a foundation for mathematics you must say:
1st - what are the basic *concepts* that must be understood without a
proper mathematical definition, and in terms of which all other
mathematical concepts are to be defined.
2nd - what are the basic *principles* (*axioms* in the old-fashioned
Euclidean sense) that must be accepted as true without proof, and which
are the propositions to which all mathematical proof ultimately appeals.
and 3rd - what methods of definition and proof are to be employed
Surely it is manifestly obvious that you cannot lay down a
genuine foundation for mathematics in this sense simply by presenting
a formal first order theory, especially if you are going to claim the
right to draw on all the facts that we have discovered about these
theories over the past 80 years or so.
We have not yet seen a "category-theoretic" account of the
foundations mathematics that is anything other than a presentation of
either a single first order theory, or a family of such theories. These
theories, described by Colin McLarty and others, have been interesting,
fundamental, important, well motivated (pace Friedman and Simpson). But
they are NOT *foundational* theories, and cannot logically be regarded
as such. And therefore they are NOT in competition with set theory as a
foundation for mathematics.
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John Mayberry
Lecturer in Mathematics
School of Mathematics
University of Bristol
J.P.Mayberry at bristol.ac.uk
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